arXiv subjects
Publications and source records attributed to Xiaodong Cao.
In this paper we develop a Li-Yau-Hamilton (LYH) type differential Harnack estimate for positive solutions to the FitzHugh-Nagumo equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including a classical Harnack inequality and a lower bound for the speed of traveling wave solutions.
We construct regularized distance functions for star-shaped domains in space forms $\mathbb N^n(K)$ and derive explicit formulas for their Hessians in terms of the principal curvatures of the boundary. As an application, we use these regularized distance functions to construct barriers and prove the existence of an admissible $C^{1,1}$ solution to the degenerate $σ_k$ equation on ring domains in $\mathbb R^n$ and $\mathbb H^n$, under suitable conditions on the boundary components.
We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp pinching condition (closely related, in a precise sense, to that of Kähler metric), then the soliton is necessarily locally Kähler. We further obtain a characterization theorem and a weighted integral gap result for compact gradient shrinking Ricci solitons with bounded modified sectional curvature. In addition, we establish a Hitchin-Thorpe type inequality for compact four-dimensional Ricci solitons, providing new topological constraints on such manifolds.
Simulating the nonequilibrium dynamics of interacting open quantum systems remains challenging beyond small system sizes. Quantum phase-space representations provide a scalable approach, but their useful simulation time can be limited by broad distribution tails and the associated boundary terms. We introduce the neural gauge-$P$ representation for open bosonic systems, in which stochastic gauges are parameterized by neural networks and optimized using exact moment equation residuals. For the driven-dissipative Bose--Hubbard model in both single-site and square-lattice settings, the neural gauge-$P$ representation remains accurate during long-time evolution toward the steady state, whereas the corresponding ungauged representation becomes unreliable at substantially earlier times. These results demonstrate the potential of the neural gauge-$P$ representation for accurate simulations of nonequilibrium open quantum many-body dynamics.
We present the combination of a complex-time tensor-network impurity solver with an analytic continuation scheme based on exponential fitting as an efficient framework for single and multi-orbital dynamical mean-field calculations. By performing time-evolution along a complex-time contour, the approach balances computational cost with the difficulty of spectral recovery, offering greater flexibility than methods confined to the real or imaginary axis. By complementing the complex-time evolution with an exponential fitting scheme, we faithfully extract real-time information at negligible cost. The resulting method obtains high-resolution spectra at a significantly lower computational cost than real-time evolution, offering a promising tool for ab initio studies of strongly correlated materials.
We uncover a dual mechanism by which oxygen defects suppress superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_{7+δ}$ using density functional theory, dynamical mean-field theory, and functional renormalization group analysis. Apical vacancies and interbilayer interstitials emerge as the dominant low-energy defect species and are further stabilized by orthorhombic domain walls. These two defect classes drive the electronic structure in opposing directions. Vacancy-induced disorder generates local magnetic moments and promotes Anderson localization at moderate concentrations, whereas periodic interstitial ordering yields a coherent but weakly correlated metallic background that fails to support superconductivity. These findings highlight the decisive role of oxygen defects in shaping the superconducting and provide microscopic guidance for improving superconductivity through controlled defect engineering.
We introduce a neural network impurity solver for real-frequency DMFT that employs a multihead cross-attention mechanism to map hybridization functions to spectral functions, conditioned on impurity parameters. Trained on high-quality MPS data from complex contour time evolution and incorporating derivative constraints with respect to the complex-time angle, our model achieves smooth generalization to the real-frequency axis. Benchmarking on the single-band Hubbard model for the Bethe lattice demonstrates quantitative accuracy across metallic, strongly correlated, and insulating regimes.
Higher-spin quantum magnets with competing interactions offer a rich platform for exploring quantum phases that transcend the paradigms of spin-1/2 systems, owing to their enlarged local Hilbert spaces and the emergence of multipolar correlations. We investigate a one-dimensional spin-1 chain where quadrupolar order is promoted by two distinct mechanisms: conventional bilinear-biquadratic exchange and bond-directional antiferromagnetic Kitaev frustration. Using density matrix renormalization group calculations, we determine the complete ground-state phase diagram and uncover two emergent phases induced by the Kitaev interaction: a Kitaev nematic phase and a Kitaev-dimer phase. The Kitaev nematic phase emerges from a fragile biquadratic dimer state via a continuous quantum phase transition in the Ising universality class. The Kitaev dimer phase spontaneously breaks a screw symmetry to favor either $x$- or $y$-spin bonding, forming a gapped state that coexists with a crystalline order of alternating $\mathbb{Z}_2$ fluxes.
Ionic conductivity is a critical property of solid ionic conductors, directly influencing the performance of energy storage devices such as batteries. However, accurately calculating ionic conductivity or diffusion coefficient remains challenging due to the complex, dynamic nature of ionic motion, which often yield significant deviations, especially at room temperature. In this study, we propose an improved method, T-MSD, to enhance the accuracy and reliability of diffusion coefficient calculations. Combining time-averaged mean square displacement analysis with block jackknife resampling, this method effectively addresses the impact of rare, anomalous diffusion events and provides robust statistical error estimates from a single simulation. Applied to large-scale deep-potential molecular dynamics simulations, we show that T-MSD eliminates the need for multiple independent simulations while ensuring accurate diffusion coefficient calculations across systems of varying sizes and simulation durations. This approach offers a practical and reliable framework for precise ionic conductivity estimation, advancing the study and design of high-performance solid ionic conductors.
In condensed matter physics, various mechanisms give rise to distinct insulating phases. The competition and interplay between these phases remain elusive, even for the seemingly most distinguishable band and Mott insulators. In multilayer systems, such interplay is mediated by interlayer hopping, which competes with the Coulomb repulsion to determine the nature of insulators. The layered compound 1T-TaS_{2} provides an ideal platform for investigating this phenomenon, as it naturally hosts coexisting Mott and band insulating states. However, distinguishing these distinct insulating states and characterizing the evolution remain challenging. In this study, we employ a dual approach utilizing surface-sensitive High-Resolution Electron Energy Loss Spectroscopy (HREELS) and bulk-sensitive Fourier-transform Infrared Spectroscopy (FTIR) to investigate the electronic excitation spectrum of 1T-TaS_{2}. Our methodology effectively identifies the features originating from the Mott and band insulators by analyzing the differences in their bulk and surface spectral weights, along with their energy distinctions. Based on the previous identification, we further investigate the evolution of insulating state features in the homostructure as they are modulated by temperature. The measurements and Dynamical Mean-Field Theory (DMFT) calculations suggest that the softening and broadening of Hubbard excitations in the Mott state with increasing temperature result from enhanced interlayer hopping between the Mott and band insulators.
The continued development of computational approaches to many-body ground-state problems in physics and chemistry calls for a consistent way to assess its overall progress. In this work, we introduce a metric of variational accuracy, the V-score, obtained from the variational energy and its variance. We provide an extensive curated dataset of variational calculations of many-body quantum systems, identifying cases where state-of-the-art numerical approaches show limited accuracy, and future algorithms or computational platforms, such as quantum computing, could provide improved accuracy. The V-score can be used as a metric to assess the progress of quantum variational methods toward a quantum advantage for ground-state problems, especially in regimes where classical verifiability is impossible.
[Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of examples; it also provides some geometric aspects of gradient Ricci solitons, including various asymptotic behaviors; finally, it discusses some recent results on classification and rigidity. In particular, this survey focuses on dimension four.
In this paper, we propose a delayed cytokine-enhanced viral infection model incorporating saturation incidence and immune response. We compute the basic reproduction numbers and introduce a convex cone to discuss the impact of non-negative initial data on solutions. By defining appropriate Lyapunov functionals and employing LaSalle's invariance principle, we investigate the stability of three equilibria: the disease-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium. We establish conditions under which these equilibria are globally asymptotically stable. Numerical analyses not only corroborate the theoretical results but also reveal that intervention in virus infection can be achieved by extending the delay period.
Transformer neural networks, known for their ability to recognize complex patterns in high-dimensional data, offer a promising framework for capturing many-body correlations in quantum systems. We employ an adapted Vision Transformer (ViT) architecture to model quantum impurity models, optimizing it with a subspace expansion scheme that surpasses conventional variational Monte Carlo in both accuracy and efficiency. Benchmarks against matrix product states in single- and three-orbital Anderson impurity models show that these ViT-based neural quantum states achieve comparable or superior accuracy with significantly fewer variational parameters. We further extend our approach to compute dynamical quantities by constructing a restricted excitation space that effectively captures relevant physical processes, yielding accurate core-level X-ray absorption spectra. These findings highlight the potential of ViT-based neural quantum states for accurate and efficient modeling of quantum impurity models.
We present an approach to tame the growth of entanglement during time evolution by tensor network methods. It combines time evolution in the complex plane with a perturbative and controlled reconstruction of correlation functions on the real-time axis. We benchmark our approach on the single impurity Anderson model. Compared to purely real-time evolution, the complex time evolution significantly reduces the required bond dimension to obtain the spectral function. Notably, our approach yields self-energy results with high precision at low frequencies, comparable to numerical renormalization group (NRG) results, and it successfully captures the exponentially small Kondo energy scale.
We present a minimally entangled typical thermal state (METTS) quantum impurity solver for general multi-orbital systems at finite temperatures. We introduce an improved estimator for the single-particle Green's function that strongly reduces the large fluctuations at long imaginary time and low temperature, which were a severe limitation of the original algorithm. In combination with the fork tensor product states ansatz, we obtain a dynamical mean field theory (DMFT) quantum impurity solver, which we benchmark for single and three-band models down to low temperatures, including the effect of spin-orbit coupling in a realistic DMFT computation for the Hund's metal Sr$_2$RuO$_4$ down to low temperatures.
The Anderson model for a magnetic impurity in a one-dimensional quasicrystal is studied using the numerical renormalization group (NRG). The main focus is elucidating the physics at the critical point of the Aubry-Andre (AA) Hamiltonian, which exhibits a fractal spectrum with multifractal wave functions, leading to an AA Anderson (AAA) impurity model with an energy-dependent hybridization function defined through the multifractal local density of states at the impurity site. We first study a class of Anderson impurity models with uniform fractal hybridization functions that the NRG can solve to arbitrarily low temperatures. Below a Kondo scale $T_K$, these models approach a fractal strong-coupling fixed point where impurity thermodynamic properties oscillate with $\log_b T$ about negative average values determined by the fractal dimension of the spectrum. The fractal dimension also enters into a power-law dependence of $T_K$ on the Kondo exchange coupling $J_K$. To treat the AAA model, we combine the NRG with the kernel polynomial method (KPM) to form an efficient approach that can treat hosts without translational symmetry down to a temperature scale set by the KPM expansion order. The aforementioned fractal strong-coupling fixed point is reached by the critical AAA model in a simplified treatment that neglects the wave-function contribution to the hybridization. The temperature-averaged properties are those expected for the numerically determined fractal dimension of $0.5$. At the AA critical point, impurity thermodynamic properties become negative and oscillatory. Under sample-averaging, the mean and median Kondo temperatures exhibit power-law dependences on $J_K$ with exponents characteristic of different fractal dimensions. We attribute these signatures to the impurity probing a distribution of fractal strong-coupling fixed points with decreasing temperature.
In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a Kähler model. The first theorem could be considered as a rigidity result for the Kähler-Ricci soliton structure on $\mathbb{S}^2\times \mathbb{R}^2$ (in the sense of Remark 1). Moreover, we show that if the quotient of norm of the self-dual Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half-conformally flat or locally Kähler.